Dependent variables in calibration curves

In calibration, concentration (or concentration times sputtering
rate) is plotted as the dependent variable (y-axis) and Intensity
is plotted as the independent variable (x-axis). Why? And
why do some text books show calibration curves the other way round?
Are they equivalent? In general, they are not equivalent.

The reason we plot Intensity as the independent variable: it is
convenient for analysis. When an analyte intensity is measured,
it can then be converted directly into concentration (or concentration
times sputtering rate) using the calibration function.

ciqM or Ii?

In GD-OES, a second order calibration function could be expressed
either as

or

And either could be transformed into the other using the algorithm
in ref (1). For example, the first equation would transform into
the second with

So, mathematically, if the coefficients of the polynomials are
known then one equation could be transformed into the other. The
problem is that the coefficients are not known, until after regression.
And, in general, the values of the regression coefficients will
depend on how the regression is carried out. Crucial to this is
the choice of the dependent variable.

Example

Consider the following calibration for Al 396 nm, made
with RF GD-OES, using a variety of matrices, after DC bias correction.
For simplicity I have subtracted the background signal.

First the calibration is plotted with cAl.qM as independent variable.

Then with IAl as independent variable.

If we transform the equation in the first graph to match the second
we get

which is significantly different from the equation shown with the
graph. This transformed equation is plotted as the dotted line in
the second graph. The two calibration curves are clearly not the
same.

Explanation

The inverse function of a 2nd order polynomial involves necessarily
a square root. The inverted 2nd order polynomial is therefore not
exactly another second order polynomial.

The exact form of GD calibration curve involving self-absorption
is not even a second order polynomial but a rather complex function.
If we can use 2nd order polynomial to fit the calibration data,
this only means that out model reproduces the data within the limits
of the uncertainty.

When calibration function rather than analytical function are used
to fit the calibration data, the inversion process just needs to
be sufficiently precise to reproduce the data within the limits
of measurement uncertainty, which may involve a higher order polynomial
for the inverted function. In general, however, a second order polynomial
will be sufficient, to reproduce the data, within the limits. For
more detailed discussion check our the references given below.

References

M Abramowitz and I A Stegun, Handbook
of Mathematical Functions with Formulas, Graphs and Mathematical
Tables, John Wiley, New York (1972), p 16.